for every closed hyperbolic surface \(S\)  of genus \(g\)  and every admissible function \(f\)  [1]}. Note that Buser's convention for the Fourier transform in [1]} is different from ours, so the formula we obtain is slightly different. If we take take \(f\)  to be the same as what Buser calls \(g\) , then the function \(h\)  in Buser's notation is equal to \(\sqrt{2 \pi }\widehat{f}\) . Equation (REF ) is obtained by dividing Buser's formula throughout by \(\sqrt{2 \pi }\)  (note that \(\operatorname{area}(S) = 4\pi (g-1)\)  and \(\int _{-\infty }^\infty r \widehat{f}(r) \tanh (\pi r) dr = 2 \int _0^\infty r \widehat{f}(r) \tanh (\pi r) dr\) ).
